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Circumscribed circle

This article is about circumscribed circles in Geometry. For the use of circumscribed in Biological classification, see Circumscription (taxonomy).

Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P

In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.

A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. Not every polygon has a circumscribed circle, as the vertices of a polygon do not need to all lie on a circle, but every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm.[2] Even if a polygon has a circumscribed circle, it may not coincide with its minimum bounding circle; for example, for an obtuse triangle, the minimum bounding circle has the longest side as diameter and does not pass through the opposite vertex.

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Construction of the circumcircle (red) and the circumcenter Q (red dot)

The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. For three non-collinear points,
these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector is equidistant from the two points that it bisects, from which it follows that this point, on both bisectors, is equidistant from all three triangle vertices.
The circumradius is the distance from it to any of the three vertices.

Alternate construction of the circumcenter (intersection of broken lines)

An alternate method to determine the circumcenter is to draw any two lines each one departing from one of the vertices at an angle with the common side, the common angle of departure being 90° minus the angle of the opposite vertex. (In the case of the opposite angle being obtuse, drawing a line at a negative angle means going outside the triangle.)

In coastal navigation, a triangle's circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies.

we then have a|v|2 − 2Sv − b = 0 and, assuming the three points were not in a line (otherwise the circumcircle is that line that can also be seen as a generalized circle with S at infinity), |v − S/a|2 = b/a + |S|2/a2, giving the circumcenter S/a and the circumradius √b/a + |S|2/a2. A similar approach allows one to deduce the equation of the circumsphere of a tetrahedron.

Hence, given the radius, r, center, Pc, a point on the circle, P0 and a unit normal of the plane containing the circle, n^{\displaystyle \scriptstyle {\hat {n}}}, one parametric equation of the circle starting from the point P0 and proceeding in a positively oriented (i.e., right-handed) sense about n^{\displaystyle \scriptstyle {\hat {n}}} is the following:

Additionally, the circumcircle of a triangle embedded in d dimensions can be found using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle. We start by transposing the system to place C at the origin:

This formula only works in three dimensions as the cross product is not defined in other dimensions, but it can be generalized to the other dimensions by replacing the cross products with following identities:

Without loss of generality this can be expressed in a simplified form after translation of the vertex A to the origin of the Cartesian coordinate systems, i.e., when A′ = A − A = (A′x,A′y) = (0,0). In this case, the coordinates of the vertices B′ = B − A and C′ = C − A represent the vectors from vertex A′ to these vertices. Observe that this trivial translation is possible for all triangles and the circumcenter U′=(Ux′,Uy′){\displaystyle U'=(U'_{x},U'_{y})} of the triangle A′B′C′ follow as

Since the Cartesian coordinates of any point are a weighted average of those of the vertices, with the weights being the point's barycentric coordinates normalized to sum to unity, the circumcenter vector can be written as

If and only if a triangle is acute (all angles smaller than a right angle), the circumcenter lies inside the triangle.

If and only if it is obtuse (has one angle bigger than a right angle), the circumcenter lies outside the triangle.

If and only if it is a right triangle, the circumcenter lies at the center of the hypotenuse. This is one form of Thales' theorem.

The circumcenter of an acute triangle is inside the triangle

The circumcenter of a right triangle is at the center of the hypotenuse

The circumcenter of an obtuse triangle is outside the triangle

These locational features can be seen by considering the trilinear or barycentric coordinates given above for the circumcenter: all three coordinates are positive for any interior point, at least one coordinate is negative for any exterior point, and one coordinate is zero and two are positive for a non-vertex point on a side of the triangle.

The angles which the circumscribed circle forms with the sides of the triangle coincide with angles at which sides meet each other. The side opposite angle α meets the circle twice: once at each end; in each case at angle α (similarly for the other two angles). This is due to the alternate segment theorem, which states that the angle between the tangent and chord equals the angle in the alternate segment.

where a, b, c are the lengths of the sides of the triangle and s = (a + b + c)/2 is the semiperimeter. The expression s(s−a)(s−b)(s−c){\displaystyle {\sqrt {\scriptstyle {s(s-a)(s-b)(s-c)}}}} above is the area of the triangle, by Heron's formula.[3] Trigonometric expressions for the diameter of the circumcircle include[4]:p.379

The useful minimum bounding circle of three points is defined either by the circumcircle (where three points are on the minimum bounding circle) or by the two points of the longest side of the triangle (where the two points define a diameter of the circle). It is common to confuse the minimum bounding circle with the circumcircle.

The circumcircle of three collinear points is the line on which the three points lie, often referred to as a circle of infinite radius. Nearly collinear points often lead to numerical instability in computation of the circumcircle.

Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius.[11]:p.83 Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle.

If a triangle has two particular circles as its circumcircle and incircle, there exist an infinite number of other triangles with the same circumcircle and incircle, with any point on the circumcircle as a vertex. (This is the n=3 case of Poncelet's porism). A necessary and sufficient condition for such triangles to exist is the above equality OI=R(R−2r).{\displaystyle OI={\sqrt {R(R-2r)}}.}[9]:p. 188

For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal, and sides 2, 4, 6, ... are equal).[12]

In any cyclic n-gon with even n, the sum of one set of alternate angles (the first, third, fifth, etc.) equals the sum of the other set of alternate angles. This can be proven by induction from the n=4 case, in each case replacing a side with three more sides and noting that these three new sides together with the old side form a quadrilateral which itself has this property; the alternate angles of the latter quadrilateral represent the additions to the alternate angle sums of the previous n-gon.

Let one n-gon be inscribed in a circle, and let another n-gon be tangential to that circle at the vertices of the first n-gon. Then from any point P on the circle, the product of the perpendicular distances from P to the sides of the first n-gon equals the product of the perpendicular distances from P to the sides of the second n-gon.[9]:p. 72

Any regular polygon is cyclic. Consider a unit circle, then circumscribe a regular triangle such that each side touches the circle. Circumscribe a circle, then circumscribe a square. Again circumscribe a circle, then circumscribe a regular 5-gon, and so on. The radii of the circumscribed circles converge to the so-called polygon circumscribing constant